3.715 \(\int \sqrt{2-\sqrt{4+\sqrt{-9+5 x}}} \, dx\)

Optimal. Leaf size=82 \[ \frac{8}{45} \left (2-\sqrt{\sqrt{5 x-9}+4}\right )^{9/2}-\frac{48}{35} \left (2-\sqrt{\sqrt{5 x-9}+4}\right )^{7/2}+\frac{64}{25} \left (2-\sqrt{\sqrt{5 x-9}+4}\right )^{5/2} \]

[Out]

(64*(2 - Sqrt[4 + Sqrt[-9 + 5*x]])^(5/2))/25 - (48*(2 - Sqrt[4 + Sqrt[-9 + 5*x]])^(7/2))/35 + (8*(2 - Sqrt[4 +
Sqrt[-9 + 5*x]])^(9/2))/45

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Rubi [A]  time = 0.0801328, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {371, 1398, 772} \[ \frac{8}{45} \left (2-\sqrt{\sqrt{5 x-9}+4}\right )^{9/2}-\frac{48}{35} \left (2-\sqrt{\sqrt{5 x-9}+4}\right )^{7/2}+\frac{64}{25} \left (2-\sqrt{\sqrt{5 x-9}+4}\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 - Sqrt[4 + Sqrt[-9 + 5*x]]],x]

[Out]

(64*(2 - Sqrt[4 + Sqrt[-9 + 5*x]])^(5/2))/25 - (48*(2 - Sqrt[4 + Sqrt[-9 + 5*x]])^(7/2))/35 + (8*(2 - Sqrt[4 +
Sqrt[-9 + 5*x]])^(9/2))/45

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 1398

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \sqrt{2-\sqrt{4+\sqrt{-9+5 x}}} \, dx &=\frac{2}{5} \operatorname{Subst}\left (\int x \sqrt{2-\sqrt{4+x}} \, dx,x,\sqrt{-9+5 x}\right )\\ &=\frac{2}{5} \operatorname{Subst}\left (\int \sqrt{2-\sqrt{x}} (-4+x) \, dx,x,4+\sqrt{-9+5 x}\right )\\ &=\frac{4}{5} \operatorname{Subst}\left (\int \sqrt{2-x} x \left (-4+x^2\right ) \, dx,x,\sqrt{4+\sqrt{-9+5 x}}\right )\\ &=\frac{4}{5} \operatorname{Subst}\left (\int \left (-8 (2-x)^{3/2}+6 (2-x)^{5/2}-(2-x)^{7/2}\right ) \, dx,x,\sqrt{4+\sqrt{-9+5 x}}\right )\\ &=\frac{64}{25} \left (2-\sqrt{4+\sqrt{-9+5 x}}\right )^{5/2}-\frac{48}{35} \left (2-\sqrt{4+\sqrt{-9+5 x}}\right )^{7/2}+\frac{8}{45} \left (2-\sqrt{4+\sqrt{-9+5 x}}\right )^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0475984, size = 57, normalized size = 0.7 \[ \frac{8 \left (2-\sqrt{\sqrt{5 x-9}+4}\right )^{5/2} \left (35 \sqrt{5 x-9}+130 \sqrt{\sqrt{5 x-9}+4}+244\right )}{1575} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 - Sqrt[4 + Sqrt[-9 + 5*x]]],x]

[Out]

(8*(2 - Sqrt[4 + Sqrt[-9 + 5*x]])^(5/2)*(244 + 35*Sqrt[-9 + 5*x] + 130*Sqrt[4 + Sqrt[-9 + 5*x]]))/1575

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Maple [A]  time = 0.012, size = 59, normalized size = 0.7 \begin{align*}{\frac{64}{25} \left ( 2-\sqrt{4+\sqrt{-9+5\,x}} \right ) ^{{\frac{5}{2}}}}-{\frac{48}{35} \left ( 2-\sqrt{4+\sqrt{-9+5\,x}} \right ) ^{{\frac{7}{2}}}}+{\frac{8}{45} \left ( 2-\sqrt{4+\sqrt{-9+5\,x}} \right ) ^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2-(4+(-9+5*x)^(1/2))^(1/2))^(1/2),x)

[Out]

64/25*(2-(4+(-9+5*x)^(1/2))^(1/2))^(5/2)-48/35*(2-(4+(-9+5*x)^(1/2))^(1/2))^(7/2)+8/45*(2-(4+(-9+5*x)^(1/2))^(
1/2))^(9/2)

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Maxima [A]  time = 0.996798, size = 78, normalized size = 0.95 \begin{align*} \frac{8}{45} \,{\left (-\sqrt{\sqrt{5 \, x - 9} + 4} + 2\right )}^{\frac{9}{2}} - \frac{48}{35} \,{\left (-\sqrt{\sqrt{5 \, x - 9} + 4} + 2\right )}^{\frac{7}{2}} + \frac{64}{25} \,{\left (-\sqrt{\sqrt{5 \, x - 9} + 4} + 2\right )}^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-(4+(-9+5*x)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

8/45*(-sqrt(sqrt(5*x - 9) + 4) + 2)^(9/2) - 48/35*(-sqrt(sqrt(5*x - 9) + 4) + 2)^(7/2) + 64/25*(-sqrt(sqrt(5*x
- 9) + 4) + 2)^(5/2)

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Fricas [A]  time = 1.43526, size = 171, normalized size = 2.09 \begin{align*} -\frac{8}{1575} \,{\left (2 \,{\left (5 \, \sqrt{5 \, x - 9} - 32\right )} \sqrt{\sqrt{5 \, x - 9} + 4} - 175 \, x - 4 \, \sqrt{5 \, x - 9} + 443\right )} \sqrt{-\sqrt{\sqrt{5 \, x - 9} + 4} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-(4+(-9+5*x)^(1/2))^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-8/1575*(2*(5*sqrt(5*x - 9) - 32)*sqrt(sqrt(5*x - 9) + 4) - 175*x - 4*sqrt(5*x - 9) + 443)*sqrt(-sqrt(sqrt(5*x
- 9) + 4) + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{2 - \sqrt{\sqrt{5 x - 9} + 4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-(4+(-9+5*x)**(1/2))**(1/2))**(1/2),x)

[Out]

Integral(sqrt(2 - sqrt(sqrt(5*x - 9) + 4)), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-(4+(-9+5*x)^(1/2))^(1/2))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError